Complexity and recurrence in Hamiltonian systems
Project responsable | Félix Schlenk |
Team member |
Agnes Gadbled
Yuri Chekanov |
Abstract |
The most interesting dynamical systems are those describing systems
without friction, and many among these can be described as
Hamiltonian systems. Since there is no friction, one can expect
that Hamiltonian systems are complicated. This is what we aim to
prove in the first part of the project for many natural Hamiltonian
systems. The origin of Hamiltonian dynamics is the study of the
motion of planets, which move - fortunately! - on (almost) closed
orbits. The search for closed orbits is therefore a fundamental and
particularly beautiful problem in this field. In the second part of
the project, we aim to show that for our class of Hamiltonian
systems, every energy surface carries infinitely many closed
orbits. We shall first try to prove this for a special but very
natural class of Hamiltonian systems. The configuration spaces we
consider are those closed manifolds whose (based or free) loop
space is complicated in the sense that the dimension of its
homology grows exponentially fast. Most closed manifolds belong to
this class. The idea is that the fast homological growth of the
configuration space Q should lead to a fast growth of the orbit
complexity and of the number of closed orbits for ALL natural
Hamiltonian flows over Q at each energy level. In technical terms,
the first project aims to show that on rationally hyperbolic
manifolds, each Reeb flow on the spherization has positive
topological entropy. This would extend famous results by Dinaburg,
Gromov and Paternain on geodesic flows. The second project aims to
show that the number of closed orbits of these Reeb flows grows
exponentially fast in time. This would extend work by Gromov on
closed geodesics, and would be a strong quantitative version of the
Weinstein conjecture for these systems. |
Keywords |
Hamiltonian Dynamics, Lagrangian submanifolds, Lagrangian tori, symplectic rigidity |
Type of project | Fundamental research project |
Research area | Mathématiques |
Method of financing | FNS - Encouragement de projets (Div. I-III) |
Status | Completed |
Start of project | 1-4-2009 |
End of project | 30-9-2010 |
Overall budget | 167'980.00 |
Additional info |
http://p3.snf.ch/projects-125352# |
Contact | Félix Schlenk |